I don't think there's actually a problem here.
Normal universe model: The universe has no bias towards supporting life
I think it only looks like there's a problem because you haven't separated this into a 'chance' hypothesis and an 'ensemble' hypothesis, such that we have three initial hypotheses.
Suppose that you are a gambler and a dealer wants you to determine whether or not zer dice are biased. The thing is, ze never lets you observe a dice roll; you're only allowed to see how the dice landed after they have been rolled. Every time ze goes to roll the dice, you have to leave the room. Over and over again, each time you enter the room, you observe that the dealer has rolled a double six. If our hypothesis space is really limited to these two hypotheses, then the probability of biased dice should skyrocket, which is, as far as I can tell, the point that you're making in the article above.
But there may be a hypothesis outside of your hypothesis space. Suppose the dealer secretly only lets you into the room if ze rolls a double six. The key here is that our observations of the dice are subject to a selection bias.
It would be really surprising to see a double six if it had been rolled after just one trial (because the probability of rolling at least one double six in one trial is ~0.027), but it would be expected if there had been many trials (because the probability of rolling at least one double six, in say, 200 trials, is ~0.996). And it seems like a sort of explanation to say to the gambler that even though the prior probability of a double six being rolled in any given trial is quite low, it's not quite so surprising to see it if there have actually been many rolls that you could not observe, since we would either see that outcome or see no outcome at all.
Does that make sense to you?
"Suppose the dealer secretly only lets you into the room if ze rolls a double six"
You seem to be proposing that we should have an alternate hypothesis:
"Our observations are filtered by the requirement of us being alive"
However, this isn't an alternate hypothesis as in both the Normal universe and the Magical universe it holds.
To make it clearer, if I an examining two hypothesises:
1) "Barrack Obama is human and he is president of the United States" 2) "Barrack Obama is human and he is not president of the United States&quo...
The Fine-tuned Universe Theory, according to Wikipedia is the belief that, "our universe is remarkably well suited for life, to a degree unlikely to happen by mere chance". It is typically used to argue that our universe must therefore be the result of Intelligent Design.
One of the most common counter-arguments to this view based on the Anthropic Principle. The argument is that if the conditions were not such that life would be possible, then we would not be able to observe this, as we would not be alive. Therefore, we shouldn't be surprised that the universe has favourable conditions.
I am going to argue that this particular application of the anthropic principle is in fact an incorrect way to deal with this problem. I'll begin first by explaining one way to deal with this problem; afterwards I will explain why the other way is incorrect.
Two model approach
We begin with two modes:
However, this is actually asking the wrong question. It is right to note that we shouldn't be surprised to observe that we survived given that it would be impossible to observe otherwise. However, if we were then informed that we lived in a normal, unbiased universe, rather than in an alternate biased universe, if the maths worked out a particular way such that it leaned heavily towards the alternate universe, then we would be surprised to learn we lived in a normal universe. In particular, we showed how this could work out above, when we examined the situation where p(we exist|normal universe) approached 0. The anthropic argument against the alternate hypothesis denies that surprise in a certain sense can occur, however, if fails to show that surprised in another, more meaningful sense can occur.
=p(we exist|normal universe)p(normal universe) + 1 - p(normal universe)
Performing Bayesian updates
Again, we'll imagine that we have a biased universe where we have 100% chance of being alive.
We will use Bayes law:
p(a|b)=p(b|a)p(a)/p(b)
Where:
a = being in a normal universe
b = we are alive
We'll also use:
p(alive) = p(alive|normal universe)p(normal universe) + p(alive|biased universe)p(biased universe)
Example 1:
Setting:
p(alive|normal universe) = 1/100
p(normal universe) = 1/2
The results are:
p(we are alive) = (1/100)*(1/2)+1*(1/2) = 101/200
p(normal universe|alive) = (1/100)*(1/2)*(200/101) = 1/101
Example 2:
Setting:
p(normal universe)=100/101
p(alive|normal universe) = 1/100
p(normal universe) = 100/101
The results are:
p(we are alive) = 100/101*1/100+1/101*1 = 2/101
p(normal universe|alive) = (1/100)*(100/101)* (101/2) = 1/2